Circumplanetary disc properties obtained from radiation hydrodynamical simulations of gas accretion by protoplanets

“Canup & Ward (2002) discuss the uncertainty in the nature of such a disc, whether it is an accretion circumplanetary disc or a spin-out disc.”

“The circumplanetary disc radius is about one-third of the Hill radius (e.g. Ayliffe & Bate 2009).”

“Simulations presented by Ayliffe & Bate (2009) suggest that 5 Note that our parametrization is different from the publicly available version of FARGO. the circumplanetary disk has a radius Rcp ≃ RH/3 and aspect ratio (H/R)cp ≃ 0.5.”

“Ayliffe & Bate (2009) have shown that the accreted gas from the disk falling onto the planet forms a circumplanetary disk with a radius 1/3 of the planet’s Hill radius.”

“The above value of rtrunc is in agreement with the point of turnover of specific angular momentum found in their two-dimensional SPH simulation, as well as in the result of Ayliffe & Bate (2009b).”

“Like the 3D calculation of this work, Ayliffe & Bate (2009a) and D’Angelo & Bodenheimer (2013) do not observe the formation of disc-like structures.”

“Material that resides beyond the gap in the circumstellar disk can be accreted by the protoplanet, but forms a circumplanetary disk because of its large angular momentum (Lubow et al. 1999; Ayliffe & Bate 2009).”

“However, the CPD structures we focus on are still present in radiation hydrodynamic runs as shown by Ayliffe & Bate (2009).”

“Previous non-isothermal simulations addressing the subdisk (e.g. Ayliffe & Bate 2009a,b; D’Angelo et al. 2003; D’Angelo & Podolak 2015; Gressel et al. 2013) have already pointed out that the CPD is hot, its inner part is optically thick, and it has a steep radial temperature profile.”

“Hydrodynamic simulations of planets embedded in circumstellar disks also can find ξ ∼ 0.3 (Ayliffe & Bate 2009).”

“…specific angular momentum begins to turn, we find the CPD extends to about half the Hill radius, which is similar to but slightly larger than the CPDs reported in other studies where values of one-third of the Hill sphere radius (Ayliffe & Bate 2009a; Machida et al. 2008; Tanigawa et al. 2012) or0.”

“Quillen & Trilling (1998), Ayliffe & Bate (2009), and Martin & Lubow (2011) obtain similar results of ∼0.3– 0.4RH, which roughly corresponds the centrifugal radius.”

“The effective viscosity for the SPH simulations of Ayliffe & Bate (2009) depends on the smoothing length of the particles which is not given, but some viscous evolution of their circumplanetary disks is likely to have occurred as well (M. R. Bate 2009, private communication).”

“The value of the disc aspect ratio from equation (4) is similar to the values obtained in the simulations by Ayliffe & Bate (2009).”

“The existence of CPDs around giant planets is also conceptually and numerically predicted (Quillen & Trilling 1998; Lubow et al. 1999; Ayliffe & Bate 2009; Ward & Canup 2010), as a consequence of angular momentum conservation similar to the formation of circumstellar disks around protostars.”

“The lower limit on the emitting area (∼0.1AU2) of the orbiting component is similar to, but smaller than, the size predicted for a circumplanetary disk around a 5 MJup planet at a distance of 13 AU from HD 100546 (Ayliffe & Bate 2009; Martin & Lubow 2011a).”

“Material that falls into the planet’s Hill sphere will spin around the planet, forming CPDs to conserve the angular momentum (Lubow et al. 1999; Ayliffe & Bate 2009; Szulágyi et al. 2016).”

“In a more evolved case, in which the circumplanetary material is filling one-third of the Hill radius as proposed for example by Ayliffe & Bate (2009) the mass of the planetary objects would be of the order of 90 M⊕.”

“The required emitting area (∼0.1 AU2) of the orbiting component can be explained by a circumplanetary disk in agreement with model predictions (e.g., Ayliffe & Bate 2009).”

“CPD models in Ayliffe & Bate (2009) and Szulágyi (2017) predict temperatures up to several thousand K, which might also produce CO bandhead emission.”

“This conclusion was later confirmed by several other studies (e.g., Machida et al. 2008; Ayliffe & Bate 2009; Martin & Lubow 2011; Tanigawa et al. 2012).”

“The disk size is roughly estimated as Rd,c ≃ RH/3 (Quillen & Trilling 1998), based on numerical simulations (Ayliffe & Bate 2009; Tanigawa et al. 2012).”

“For a Jupiter-mass planet, Ayliffe & Bate (2009b) found 4500 K peak temperature in the circumplanetary disk if they assumed realistic planetary radius.”

“In Ayliffe & Bate (2009a) we compared the accretion rates achieved in models using different core radii and found that for high opacities the accretion rates did not depend significantly on the core radius that was used, but for low opacities (1 per cent and 0.1 per cent interstellar grain…”

“Actually, most of the gas that reaches the CPD comes from a vertical direction, which is perpendicular to the orbital plane (Bate et al., 2003; Machida et al., 2008; Ayliffe and Bate, 2009a,b; Tanigawa et al., 2012; Szulagyi et al., 2014).”

“Giant planets accrete gas first by the slow collapse of quasi-hydrostatic envelopes (Ikoma et al 2000, Machida et al 2010) and later by viscous transport through a circumplanetary disk (Ayliffe & Bate 2009; Uribe et al 2013).”

“…grew toward its full mass, it opened a gap in the protosolar nebula, across which gas and coupled dust continued to flow, forming the CPD (e.g., Ayliffe and Bate 2009; Tanigawa et al. 2012; Szulágyi et al. 2016; Schulik et al. 2020), which in turn captured larger heliocentric solids…”

“Theoretical models of early-stage circumplanetary disks find typical “quiescent” accretion levels between Ṁ = 10−10 M year −1 and 10−8 M year −1 (Ayliffe & Bate 2009; Lubow & Martin 2012).”

“Hydrodynamic simulations of gas giant formation (e.g., Lubow et al. 1999; Tanigawa & Watanabe 2002; Ayliffe & Bate 2009b,a; Machida et al. 2006, 2008, 2010) have shown that circumplanetary disks form during the accretion phases of giant planets.”

“While there is still ongoing gas accretion to the forming giant planet, it is surrounded by its own disk, the so-called circumplanetary disk (CPD; e.g. Quillen & Trilling 1998; Lubow, Seibert, & Artymowicz 1999; Kley 1999; Lubow & D’Angelo 2006; Ayliffe & Bate 2009, 2012.”

“…is much smaller than one-third of the Hill radius (14–20 au, assuming the companions are on a circular orbit), which is the distance at which the disks are suspected to be truncated due to tidal interactions with the primary star (Ayliffe & Bate 2009; Martin & Lubow 2011; Shabram & Boley 2013).”

“…the assumption is quite reasonable since the minimal-mass solar nebula shows a power-law density distribution (Weidenschilling 1977) and numerical simulations of circumstellar disks also show a power-law density distribution (e.g., Ayliffe & Bate 2009), it has no fundamental physics behind it.”

“Furthermore, hydrodynamic models of giant planet formation (e.g., Ayliffe & Bate 2009) make predictions of the disk radii, scale heights, and mass distributions that could be tested with spatially resolved observations, and wide companions also provide context for observations of close-in planets…”

“A combination of analytics (Quillen & Trilling 1998; Canup & Ward 2002; Ward & Canup 2010; Martin & Lubow 2011) and simulations (D’Angelo et al. 2002; Machida 2008; Ayliffe & Bate 2009; Martin & Lubow 2011) have been used to explore the morphology and evolution of circumplanetary disks.”

“Models of giant planet formation make testable predictions about the size, scale height, and mass distribution of these circumplanetary disks (e.g. Ayliffe & Bate 2009).”

“For a Jupiter-mass planet, Ayliffe & Bate (2009b) argued for T = 1600 K at the planet surface (defined at 0.02 RHill).”

“…angular momentum can not be seen from Figure 12 as the x-axes are only up to 1.5 AU, so the component from the outer part is not shown. c© 0000 RAS, MNRAS 000, 000–000 one third of the Hill radius, which is the maximal disc extent found in previous works (Ayliffe & Bate 2009; Martin & Lubow 2011).”

“Material that resides beyond the Hill sphere can still flow into the Hill sphere, but forms a circumplanetary disk (CPD) around the protoplanet to conserve angular momentum (Lubow et al. 1999; Ayliffe & Bate 2009).”

“…that we obtain for a 10 M⊕ protoplanet, modelled using RHD in an interstellar grain opacity disc, is of similar order to its mass doubling time, as found in Ayliffe & Bate (2009a), whilst for a 33 M⊕ protoplanet the migration timescale is an order of magnitude longer than the mass doubling time.”

“Gas inflow onto sBHs preferentially proceeds from the direction perpendicular to the AGN disk plane, as found in high-resolution threedimensional hydrodynamical simulations (Ayliffe & Bate 2009; Tanigawa et al. 2012; Szulágyi et al. 2022).”

“50% RHill, e.g. Szulágyi et al. 2014, Shabram & Boley 2013, Ayliffe & Bate 2009b), therefore resolving the Hill-sphere for any of our planetary masses is not possible.”

“Simulations have shown that they tend to be compact because they are tidally truncated at ∼1/3 of the Hill radius (e.g., Quillen & Trilling 1998; Ayliffe & Bate 2009; Shabram & Boley 2013).”

“Infalling gas has too much angular momentum to fall directly onto the contracted planet, and so an accretion disk, the circumplanetary disk, forms around the planet (Lunine & Stevenson 1982; Ayliffe & Bate 2009a).”

“…some special techniques, such as nestedgrid methods in Eulerian codes (Machida et al. 2008, 2010; Paardekooper & Mellema 2008; Tanigawa et al. 2012; Gressel et al. 2013; Szulagyi et al. 2014) or SPH methods (Ayliffe & Bate 2009), in addition to the recent development of high-performance computers.”

“Gressel et al. (2013) found such thick discs for masses Mp ∼ 0.3 MJ and Ayliffe & Bate (2009) found thinner discs with an aspect ratio .”

“…strong assumptions regarding the extent of the circumplanetary material, and that it could be expected that it does not entirely fill the Hill radius of the planet (Quillen & Trilling 1998; Ayliffe & Bate 2009), and consequently, the mass derived here for the perturber is only a lower mass limit.”

“Whether these conditions present for planetary mass objects, that are cooler than stars, is still subject of current investi- Corresponding author: J. Szulágyi judit.szulagyi@uzh.ch gations (Ayliffe & Bate 2009a,b; Szulágyi 2017).”

“Models of gas giant formation indicate that during the gas accretion phase, a circumplanetary disk roughly a third the size of the Hill sphere forms (Quillen & Trilling 1998; Ayliffe & Bate 2009ab; Martin & Lubow 2011a).”

“If this were a circumplanetary disk, the corresponding Hill’s radius (RHill) of the planet would be expected to be about three times larger, following models by e.g. Shabram & Boley (2013), Ayliffe & Bate (2009) and Quillen & Trilling (1998).”

“%) of the Hill radius RH (e.g., Quillen & Trilling 1998; D’Angelo et al. 2003; Ayliffe & Bate 2009; Szulágyi et al. 2014), where RH = aP (MP/3M?)”

“A similar outer disk radius was found in numerical simulations which consider only the angular momentum content of accreting gas (Ayliffe & Bate 2009).”

“Depending on the mass of the core, these vertical motions can reach supersonic velocities before shocking on the densest parts of the envelope (Ayliffe & Bate 2009b; Szulágyi & Mordasini 2017).”

“The data points presented here differ quantitatively from the classic results in Ayliffe & Bate (2009b) mainly in that our accretion curve is flatter towards the low masses and we do not see a strong flattening of accretion rates for the high masses.”

“…outer edges of circumplanetary disks are not well known, but estimates place them somewhere between 0.3 and 0.5 Hill radii (Quillen & Trilling 1998; Ayliffe & Bate 2009, 2012; Machida 2009; Ward & Canup 2010; Martin & Lubow 2011; Tanigawa et al. 2012; Szulágyi et al. 2014; Zhu et al. 2016);…”

“Ayliffe & Bate (2009a) used radiation-hydrodynamics numerical simulations to study the planet–disk system, and derived important properties such as the power law dependence of density with radius, and the scale height of the stellar disk.”

“The planetary mass threshold below which gaseous CPD formation—hence satellite formation—cannot occur is still unknown (Ayliffe & Bate 2009).”

“Martin & Lubow (2011) suggest a tidal truncation limit of ξ ∼ 0.4 and hydrodynamic simulations of planets in circumstellar disks find ξ ∼ 0.3 (Quillen & Trilling 1998; Ayliffe & Bate 2009).”

“Physically, these disks are expected to be quite thick (h/r > 0.2), possess little flaring compared to their circum-stellar analogs, and be tidally truncated to ∼1/3 RHill from gravitational influence of the host star (Ayliffe & Bate 2009; Martin & Lubow 2011; Shabram & Boley 2013).”

“…many models of such disk were built in the past (see, e.g., Lubow et al. 1999; Canup & Ward 2002; Mosqueira & Estrada 2003; Alibert et al. 2005; Ayliffe & Bate 2009; Shabram & Boley 2013; Fujii et al. 2014, 2017; Szulágyi et al. 2016, 2017), formation of the Galilean satellites is still a…”

“According to Ayliffe & Bate (2009), we could estimate the size of the secondary’s accretion radius as its centrifugal radius and find Racc,s ≈ RH/3, MNRAS 000, 1–11 (2016) where RH is the Hill radius of the black hole.”

“…of the planetary atmosphere is matched to either the Hill Ra-Corresponding dius or Bondi Radius, whichever is smaller (e.g., Pollack et al. 1996; Ayliffe & Bate 2009a; Tanigawa et al. 2012; Ormel et al. 2015; Cimerman et al. 2017; Kurokawa & Tanigawa 2018; Lambrechts & Lega 2017; B´ethune &…”

“For r R?, T (r) ∝ r−3/4, which is similar to the profile seen in some simulated circumplanetary disks (e.g., Ayliffe & Bate 2009).”

“Once the gap has opened, simulations suggest that the incoming accretion streams possess enough angular momentum to form a circumplanetary disc (e.g. D’Angelo et al. 2002; Ayliffe & Bate 2009; Martin & Lubow 2011; Szulágyi et al. 2014).”

“They varied the initial CPD profiles and used quite different models than what recent hydrodynamic models on the CPD predict (e.g. Ayliffe & Bate 2009, Tanigawa et al. 2012, Szulágyi et al. 2014, Szulágyi 2017).”

“We stress that even after envelope collapse continued inflow onto the disk may keep it too hot for the concurrent accretion of clse-in, icy satellites (see Estrada et al. (2009) and Ayliffe & Bate (2009) for further details).”

“As shown in simulations (Ayliffe & Bate 2009; Tanigawa et al. 2012), a nearly Keplerian-rotating circumplanetary disk forms around the planet; the majority of the mass is concentrated in the outer region of the disk, mainly around the circularization radius rcir of the accreted gas (see below).”

“Recent modeling indicates that circumplanetary structures are vertically extended and intrinsically 3D, although they retain the rotationally supported disk found in 2D simulations (e.g., Ayliffe & Bate 2009; Machida et al. 2010; Tanigawa et al. 2012; Gressel et al. 2013; Szulágyi et al. 2014).”

“…is faint in radio continuum, Rab et al. (2019) find that the gas disk may still extend to 1/3 Hill radius, which is the characteristic scale of companion disks under dynamical interaction (e.g., Ayliffe & Bate 2009) and is typically &10 AU for wide-orbit PMCs at hundreds of AU from their hosts.”

“…multi-dimensional simu- lations suggest that atmospheric circulation and inefficient radiative cooling may reduce accretion rates significantly (Ayliffe & Bate 2009b,a; Ormel et al. 2015; Szulágyi et al. 2016; Szulágyi & Mordasini 2017; Szulágyi 2017; Cimerman et al. 2017; Lambrechts &…”

“This has been confirmed by numerical simulations (e.g. Ayliffe & Bate 2009a,b).”

“How much planetesimal mass fragments following the collisional cascade and is delivered depends on the planetesimal strength (Charnoz and Morbidelli 2003).”

“When invoking the tidal truncation argument (e.g., Quillen & Trilling 1998; Ayliffe & Bate 2009) to estimate PMC disk sizes, which gives an expected disk size of ∼1/3 Hill radius, this results in a large radius of ∼5–60 au for 10–20 MJup PMCs at 100–1000 au from a 1 M host star.”

“0.4 (e.g. Quillen & Trilling 1998; Ayliffe & Bate 2009a; Martin & Lubow 2011; Lehébel & Tiscareno 2015).”

“…et al. 1999; D’Angelo et al. 2002; Tanigawa & Watanabe 2002; D’Angelo et al. 2003; Bate et al. 2003; Machida et al. 2008; Machida 2009; Ayliffe & Bate 2009b; Machida et al. 2010; Ward & Canup 2010; Martin & Lubow 2011; Tanigawa et al. 2012; Ayliffe & Bate 2012; Uribe et al. 2013).”

“Most theoretical studies focussed on early evolutionary stages during which the CPD is still embedded in their parent protoplanetary disks (e.g. Ayliffe & Bate 2009; Shabram & Boley 2013; Pérez et al. 2015; Szulágyi et al. 2018a) and most of these works only considered orbits of tens of au.”

“…planet mass above 1 MJ may result in the disc large enough but its elongation along the azimuthal direction by a factor of 4 contradicts numerical simulations of circumplanetary discs which show no such elongation (Ayliffe & Bate 2009; Wang et al. 2014; Ormel et al. 2015a; Szulágyi et al. 2014).”

“…formation indicate that the disk feeding the gas giant planet will have a radius roughly one-third of the Hill Sphere (Quillen & Trilling 1998; Ayliffe & Bate 2009ab; Martin & Lubow 2011), though the full accretion envelope is likely intrinsically three dimensional with complicated flow…”